2.1 Blood Oxygen Saturation and its Measurement
Oximetry is an optical method for measuring oxygen saturation in blood. Oximetry is based on the ability of different forms of haemoglobin to absorb light of different wavelengths. Oxygenated haemoglobin (HbO2) absorbs light in the red spectrum and deoxygenated or reduced haemoglobin (RHb) absorbs light in the near-infrared spectrum. When red and infrared light is passed through a blood vessel the transmission of each wavelength is inversely proportional to the concentration of HbO2 and RHb in the blood. Pulse oximeters can differentiate the alternating light input from arterial pulsing from the constant level contribution of the veins and other non-pulsatile elements. Only the alternating light input is selected for analysis. Pulse oximetry has been shown to be a highly accurate technique. Modern pulse oximeter devices aim to measure the actual oxygen saturation of the blood (SaO2) by interrogating the red and infrared PPG signals. This measurement is denoted SpO2. The aim of modem device manufacturers is to achieve the best correlation between the pulse oximeter measurement given by the device and the actual blood oxygen saturation of the patient. It is known to those skilled in the art that in current devices a ratio derived from the photoplethysmogram (PPG) signals acquired at the patients body is used to determine the oxygen saturation measurement using a look up table containing a pluracy of corresponding ratio and saturation values. Modern pulse oximeter devices also measure patient heart rate. Current devices do not provide a measure of respiration directly from the PPG signal. Additional expensive and obtrusive equipment is necessary to obtain this measurement.
2.2 Time-Frequency Analysis in Wavelet Space
The wavelet transform of a signal x(t) is defined as
                              T          ⁡                      (                          a              ,              b                        )                          =                              1                          a                                ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                          x                ⁡                                  (                  t                  )                                            ⁢                                                ψ                  *                                ⁡                                  (                                                            t                      -                      b                                        a                                    )                                            ⁢                                                          ⁢                              ⅆ                t                                                                        [        1        ]            where ψ*(t) is the complex conjugate of the wavelet function ψ(t), a is the dilation parameter of the wavelet and b is the location parameter of the wavelet. The transform given by equation (1) can be used to construct a representation of a signal on a transform surface. The transform may be regarded as a time-scale representation or a time-frequency representation where the characteristic frequency associated with the wavelet is inversely proportional to the scale a. In the following discussion ‘time-scale’ and ‘time-frequency’ may be interchanged. The underlying mathematical detail required for the implementation within a time-scale or time-frequency framework can be found in the general literature, e.g. the text by Addison (2002).
The energy density function of the wavelet transform, the scalogram, is defined asS(a,b)=|T(a,b)|2  [2]where ‘| |’ is the modulus operator. The scalogram may be rescaled for useful purpose. One common repealing is defined as
                                          S            R                    ⁡                      (                          a              ,              b                        )                          =                                                                          T                ⁡                                  (                                      a                    ,                    b                                    )                                                                    2                    a                                    [        3        ]            and is useful for defining ridges in wavelet space when, for example, the Morlet wavelet is used. Ridges are defined as the locus of points of local maxima in the plane. Any reasonable definition of a ridge may be employed in the method. We also include as a definition of a ridge herein paths displaced from the locus of the local maxima. A ridge associated with only the locus of points of local maxima in the plane we label a ‘maxima ridge’. For practical implementation requiring fast numerical computation the wavelet transform may be expressed in Fourier space and the Fast Fourier Transform (FFT) algorithm employed. However, for a real time application the temporal domain convolution expressed by equation (1) may be more appropriate. In the discussion of the technology which follows herein, the ‘scalogram’ may be taken to the include all reasonable forms of rescaling including but not limited to the original unscaled wavelet representation, linear rescaling and any power of the modulus of the wavelet transform may be used in the definition.
As described above the time-scale representation of equation (1) may be converted to a time-frequency representation. To achieve this, we must convert from the wavelet a scale (which can be interpreted as a representative temporal period) to a characteristic frequency of the wavelet function. The characteristic frequency associated with a wavelet of arbitrary a scale is given by
                    f        =                              f            c                    a                                    [        4        ]            where fc, the characteristic frequency of the mother wavelet (i.e. at a=1), becomes a scaling constant and f is the representative or characteristic frequency for the wavelet at arbitrary scale a.
Any suitable wavelet function may be used in the method described herein. One of the most commonly used complex wavelets, the Morlet wavelet, is defined as:ψ(t)=π−3/4(ef2πf0f−e−(2πf0f 3/2)e−t3/2  [5]where f0 is the central frequency of the mother wavelet. The second term in the brackets is known as the correction term, as it corrects for the non-zero mean of the complex sinusoid within the Gaussian window. In practice it becomes negligible for values of f0>>0 and can be ignored, in which case, the Morlet wavelet can be written in a simpler form as
                              ψ          ⁡                      (            t            )                          =                              1                          π                              1                /                4                                              ⁢                      ⅇ                          ⅈ              ⁢                                                          ⁢              2              ⁢              π              ⁢                                                          ⁢                              f                0                            ⁢              t                                ⁢                      ⅇ                                          -                                  t                  3                                            /              2                                                          [        6        ]            
This wavelet is simply a complex wave within a Gaussian envelope. We include both definitions of the Morlet wavelet in our discussion here. However, note that the function of equation (6) is not strictly a wavelet as it has a non-zero mean, i.e. the zero frequency term of its corresponding energy spectrum is non-zero and hence it is inadmissible. However, it will be recognised by those skilled in the art that it can be used in practice with f0>>0 with minimal error and we include it and other similar near wavelet functions in our definition of a wavelet herein. A more detailed overview of the underlying wavelet theory, including the definition of a wavelet function, can be found in the general literature, e.g. the text by Addison (2002). Herein we show how wavelet transform features may be extracted from the wavelet decomposition of pulse oximeter signals and used to provide a range of clinically useful information within a medical device.